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Written in an episodic, snapshot manner, the novel has 155 chapters, the last 99 designated as "expendable." Some of these "expendable" chapters fill in gaps that occur in the main storyline, while others add information about the characters or record the aesthetic or literary speculations of a writer named Morelli who makes a brief appearance in the narrative. Some of the "expendable" chapters at first seem like random musings, but upon closer inspection solve questions that arise during the reading of the first two parts of the book.

An author's note suggests that the book would best be read in one of two possible ways, either progressively from chapters 1 to 56 or by "hopscotching" through the entire set of 155 chapters according to a "Table of Instructions" designated by the author. Cortázar also leaves the reader the option of choosing a unique path through the narrative.

Several narrative techniques are employed throughout the book, and frequently overlap, including first person, third person, and a kind of stream-of-consciousness. Traditional spelling and grammatical rules are often bent and sometimes broken outright.

Hopscotch is an account of the life of Horacio Oliveira, an Argentinean intellectual. He experiences life in Paris in the 1950s. The other characters consist of La Maga and a band of bohemian intellectuals who call themselves the Serpent Club.

The story opens with Horacio searching the bridges of Paris for La Maga, who has disappeared. The story progresses in a non-linear order. Scenes with the Serpent Club, who convene in the evening to drink, listen to jazz music, and engage in dialectical arguments about art and life take place between scenes of Oliveira's solitary jaunts through the city.

La Maga disappears. Horacio and his friend Etienne visit an old man who Horacio had witnessed being struck by a car on one of his walks alone through Paris. He turns out to be Morelli, an iconoclastic writer and literary critic much beloved of the Serpent Club. That night, the Club goes to Morelli's flat to assemble his new work, a novel that can be read in any order. There, a member of the Club, Babs, attacks Horacio for what he did to La Maga. The Club disbands. Horacio retreats to a bridge, where he meets a homeless woman, Emmanuele. The police arrest Emmanuele and Oliveira for lewd behavior.

Horacio returns to Buenos Aires, stopping at La Maga's hometown of Montevideo on the way. He is met in Argentina by his old friend Manolo Traveler, a worker at a circus who has never left the country. Traveler is married to a former pharmacist, Talita. He begins work as a traveling gabardine salesman, and moves in with an old girlfriend, Gekrepten, in the same hotel as the Traveler and Talita. Horacio resolves to spend more time with them, to observe their seemingly happy marriage.

Traveler gets Horacio a job with the circus, and they begin to spend most of their time together. The strong bond between Horacio and Traveler begins to put a strain on the Traveler's marriage, and Talita begins to suspect a romantic tension growing between her and Horacio.

Eventually, the owner of the circus sells it to buy a mental asylum. Traveler, Talita, and Horacio all move to the hospital to begin new jobs. While there, Horacio begins to confuse Talita with La Maga. One night, he sees her playing hopscotch in the courtyard – a game that has come to represent for him his search for an unattainable contentment. He talks to her for the first time about La Maga, and the pity she shows for him moves him to kiss her. The next morning, Traveler goes to Horacio's room to talk to him. Horacio threatens to kill himself, causing a commotion in the hospital. Traveler talks to Horacio about their relationship and Horacio's mental stability, but in the end, Horacio jumps. He survives, but all three employees are fired.[3]

The second part of the book is entitled, "From this Side," and the action takes place in Argentina. It opens with a brief introduction to the life of Manolo Traveler, Horacio's friend from childhood, who lives in Buenos Aires with his wife Talita. Although Traveler is a restless individual, the marriage appears to be on solid ground until Traveler is informed, through Horacio's old Argentinian lover, Gekrepten, that Oliveira, whom he hasn't seen in many years, is due to arrive by boat. The news fills him with a dark sense of foreboding; nonetheless, he and Talita greet Horacio at the docks, where Oliveira momentarily mistakes Talita for La Maga. Horacio then settles with Gekrepten in a hotel room that is located directly across the street from the flat Traveler and Talita share, where his mind slowly begins to unravel.

Traveler and Talita work for a circus, and when Horacio’s temporary employment as a seller of fabrics falls through, Traveler arranges for his old friend to be hired on there as well, though not without misgivings. Oliveira's presence has begun to disturb him, but he is unable to determine why. He wishes to ascribe it to Horacio's flirtations with Talita, but cannot do so, as there seems to be something more going on. And anyway, he has no doubts about his wife's remaining faithful to him. Unable to decipher the mystery, and unable to tell Horacio to leave them alone, he begins to sleep less and less, and his sense of restlessness increases.

Horacio, meanwhile, observing the relationship between Traveler and Talita, who more and more reminds him of La Maga, endeavors to enter more intimately into their lives, but he is unable to do so. His frustrations increase, and he begins to show signs of an impending mental breakdown. One hot afternoon, he spends hours on the floor trying to straighten nails, although he has no particular use in mind for them. This symbolic act then spirals out of control when he convinces Traveler and Talita to try to build a makeshift bridge with him between the windows of the two buildings over which Talita can cross. Horacio tells Talita to bring him straight nails and some yerba. Traveler indulges his friend's eccentric behavior, but Talita is frightened by the proceedings and must be prodded into participating. She feels it is a test of some sort. In the end, she tosses the yerba and nails to Oliveira without crossing over.

Soon after this incident, the owner of the circus sells the operation to a Brazilian businessman and invests in a local mental institution. Traveler, Talita and Horacio decide to go to work there despite the irony of the situation, or perhaps because of it. Horacio jests that the patients in the hospital will be no more mad than the three of them, anyway. On the day the ownership of the hospital is to be transferred, they are told that all the inmates must agree to the deal by signing a document, and that the three of them must act as witnesses. They meet a good-natured orderly named Remorino as well as a Dr. Ovejero, who manages the facility. The former owner of the circus and his wife, Cuca, are also present. One by one the inmates are led into the room where the document is to be signed, a procedure that lasts well into the night. The patients are usually referred to by their room numbers rather than their names, and they demonstrate mostly placid natures.

Talita becomes the resident pharmacist at the hospital, while Horacio and Traveler act as either orderlies or guards at night. The place is dark and eerie in the long hours before dawn, with the three often seeking refuge in alcohol and conversation in the pharmacy's warmer atmosphere. Remorino shows Horacio and Traveler the basement, where dead bodies are kept and cold beer can be had.

One night Horacio is smoking in his room when he sees Talita crossing the moonlit garden below, apparently heading to bed. A moment later, he thinks he sees La Maga appear and begin a game of hopscotch in the same general area; but when she looks up at him, he realizes it is Talita, who had turned and recrossed the garden. A kind of guilt, fed in part by the institution's gloomy atmosphere, begins to steal into his musings, and it is not long before he conceives of the idea of someone's trying to murder him while he is on duty—perhaps Traveler.

Later in the night, while Oliveira is on the second floor pondering over the symbolic implications of the mental institution's elevator, Talita approaches and the two talk about holes, passages, pits, and La Maga, of course, and as they do, the elevator comes to life, ascending from the basement. One of the mental patients is inside. After sending the man back to his room, Horacio and Talita decide to go down, ostensibly to see what he was up to.

Alone with Talita and the dead bodies, Horacio finds himself talking to her not as if she reminded him of La Maga, but as if she were La Maga. In a final moment of desperation, he attempts to kiss this Talita/La Maga mixture, but is repelled. Returning to her room, Talita tells Traveler about it, who surmises that something may be seriously wrong with his friend.

Meanwhile, having retreated to his own room, Horacio is now convinced that Traveler is coming to kill him. He begins to construct a kind of defense line in the dark room that is intended to confuse and irritate an attack, rather than deter it: water-filled basins placed on the floor, for example, as well as threads tied to heavy objects (which are in turn tied to the doorknob).

Horacio then sits in the dark on the opposite side of the room, near the window, waiting to see what will happen. The hours pass slowly and painfully, but finally Traveler does try to come in, and the tumult that results brings Dr. Ovejero and the others out into the garden, where they find Oliveira leaning out the window of his room as if intending to let himself fall. Traveler tries to talk Horacio out of doing what he, for his part, insists he doesn't intend to do, though at the conclusion of this part of the book, he suddenly muses that maybe he does mean to do such a thing after all, that maybe it is for the best, and the end of the passage is wholly open to this interpretation. Only by proceeding to read the Expendable Chapters will the reader be able to place Horacio firmly back inside mental institution, where, after being sedated by Ovejero, he succumbs to a lengthy delirium.

The third section of the book does not need to be read in order to understand the plot, but it does contain solutions to certain puzzles that arise during the perusal of the first two parts. For example, the reader finds out a great deal more about the mysterious Morelli, as well as finding out how La Maga and Emmanuele first became acquainted. Through Morelli's writings, Cortazar hints at some of the motives behind the actual construction of Hopscotch (such as a desire to write a work in which the reader is a true co-conspirator). The inner workings of Horacio Oliveira himself are described in a much less evasive manner than in any previous chapters, as well. The section, and the book, ends with Horacio visiting Morelli in the hospital, who asks him to go to his apartment and organize his notebooks while he is away. Most of these notebooks are unpublished and Oliveira not only considers doing this work as a great honor to himself personally, but also as perhaps the best chance yet of his attaining the ninth square in his lifelong game of spiritual, emotional, and metaphysical hopscotch.

The main character, Horacio Oliveira, is a well-read and loquacious bohemian. He enjoys a mostly intellectual participation in life instead of pursuing an active role [clarification needed] and appears to be obsessed with attaining what is referred to as a unifying conception of life, or center in which he can contentedly exist.

"La Maga" (a nickname, as her real name is Lucía) is a beguiling, intelligent being whose love of life and spontaneous nature challenge Horacio's ego as well as his assumptions about life. Oliveira's lover in Paris, she is lively, an active participant in her own adventures, and a stark contrast to Horacio's other friends, with whom he has formed a philosophical social circle called the Serpent Club. She eventually develops into an indispensable muse for Horacio and a lens he employs to examine himself and the world in a more three dimensional manner. La Maga also has an infant son, Rocamadour, whose appearance in France causes a crisis in the relationship between Lucía and Oliveira.

Apart from Horacio and La Maga, the other members of the Serpent Club include Ossip Gregorovius, who is presented as a rival for Lucía's affections, the artists Perico Romero and Etienne, Etienne's friend Guy Monod, Wong, and Ronald and Babs (who are married). The club meets either in La Maga's apartment or the flat Ronald and Babs share.

When Horacio returns to Argentina he is greeted by his old friend Manolo, nicknamed "Traveler," and his wife, Talita. The two are employed at a circus and seem to enjoy a mostly serene existence. Talita bears a striking similarity to Horacio's great love interest, La Maga, while Traveler is referred to as his "doppelgänger." The relationship that develops between the three centers around Oliveira's interest in Talita, which seems disengenuous, and Traveler's attempts to lessen Horacio's metaphysical burden. Oliveira desires to inhabit Traveler's life, while Traveler appears to be mostly concerned with his friend's deteriorating mental health.

Other major characters include Pola, another Parisian love interest of Horacio's, who is diagnosed with breast cancer after being cursed by La Maga; Gekrepten, Horacio's lover in Argentina; and Morelli, an Italian writer much discussed by the Serpent Club. Minor characters of note include Madame Berthe Trepat, a composer who mistakes Horacio's attention for sexual interest, and a homeless woman named Emmanuele with whom Oliveira has a brief, disastrous tryst with shortly after La Maga's disappearance.

Horacio says of himself, "I imposed the false order that hides the chaos, pretending that I was dedicated to a profound existence while all the time it was the one that barely dipped its toe into the terrible waters" (end of Chapter 21). Horacio's life follows this description as he switches countries, jobs, and lovers. The novel also attempts to resemble order while ultimately consisting of chaos. It possesses a beginning and an end but traveling from one to the other seems to be a random process. Horacio's fate is just as vague to the reader as it is to him. The same idea is perfectly expressed in improvisational jazz. Over several measures, melodies are randomly constructed by following loose musical rules. Cortázar does the same by using a loose form of prose, rich in metaphor and slang, to describe life.

Horacio drifts from city to city, job to job, love to love, life to life, yet even in his nomadic existence he tries to find a sense of order in the world’s chaos. He is always isolated: When he is with La Maga, he cannot relate to her; when he is with the Club, he is superior; when he is with Traveler and Talita, he fights their way of life. Even when with Morelli, the character he relates to most, there exists the social barriers of patient and orderly. Order versus chaos also exists in the structure of the novel, as in Morelli’s statement, "You can read my book any way you want to” (556). At the end of chapter 56, he realizes that he is neither on 'the territory' (Traveler's side, with society) nor on 'the bedroom' (what would be his side, his real place, if he had reached it).

One of the biggest arguments between Horacio and Ossip, one that threatens to put a rift in the club, is what Horacio deems “the conundrum of consciousness” (99). Does art prove consciousness? Or is it simply a continuation of instinctual leanings toward the collective brain? Talita argues a similar point in her seesaw-questions game with Horacio, who believes that only when one lives in the abstract and lets go of biological history can one achieve true consciousness.

Horacio’s life seems hopeless because he has deemed himself a failure. La Maga’s life seems hopeless because she has never worked to resolve the issues of rape and abuse in her childhood. Traveler’s life seems hopeless because he has never done what he wanted to do, and even the name he’s adopted teases at this irony. But none of these people are considered by outward society to be failures. They are stuck where they are because of their own self-defeating attitudes.

Short chapters also express the idea that there is no penetrating purpose to the novel and life in general. For Horacio, life is a series of artistic flashes by which he perceives the world in profound ways but still remains unable to create anything of value. Other major themes include obsession, madness, life-as-a-circus, the nature and meaning of sex, and self-knowledge.